Almost Quaternion-Hermitian Manifolds

Abstract

Following the point ofview of Gray and Hervella, we derive detailed conditions whichcharacterize each one of the classes of almostquaternion-Hermitian 4n-manifolds, n > 1. Previously, bycompleting a basic result of Swann, we give explicitdescriptions of the tensors contained in the space of covariantderivatives of the fundamental form Ω and split thecoderivative of Ω into its Sp(n)Sp(1)-components. For 4n > 8, Swann also proved that all theinformation about the intrinsic torsion ∇Ω iscontained in the exterior derivative dΩ. Thus, wegive alternative conditions, expressed in terms of dΩ, to characterize the different classes of almostquaternion-Hermitian manifolds.